Convert all complex numbers to trigonometric form and then simplify the expression. Write the answer in standard form.? $\frac{{(2 + 2 i)}^{5} {(- 3 + i)}^{3}}{{(\sqrt{3} + i)}^{10}}$ $((2+2i)^5 (-3+i)^3) /(sqrt 3+i)^10$

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Convert all complex numbers to trigonometric form and then simplify the expression. Write the answer in standard form.? $\frac{{(2 + 2 i)}^{5} {(- 3 + i)}^{3}}{{(\sqrt{3} + i)}^{10}}$ $((2+2i)^5 (-3+i)^3) /(sqrt 3+i)^10$

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Answer 1 · 2018-05-14T11:40:23

$\frac{27 \sqrt{3}}{4} - i \frac{27}{4}$ $(27sqrt3)/4-i27/4$

Explanation:

So this took me so many attempts and here it goes:

Find the r values using this formula:

$r = \sqrt{x^{2} + y^{2}}$ $r=sqrt (x^2+y^2)$

Plug in:

$\frac{{[2 \sqrt{2} ({cos 45}^{o} + i {sin 45}^{o})]}^{5} {[3 \sqrt{2} ({cos 135}^{o} + i {sin 135}^{o})]}^{3}}{{[2 ({cos 30}^{o} + i {sin 30}^{o})]}^{10}}$ $([2sqrt2 (cos45^o + isin45^o)]^5[3sqrt2 (cos135^o + isin135^o)]^3 )/[2(cos 30^o + isin30^o)]^10$

Power and Multiply:

${(2 \sqrt{2})}^{5}$ $(2sqrt2)^5$ see how I powered the $2 \sqrt{2}$ $2sqrt2$ and you do the same with the rest

$\frac{{(2 \sqrt{2})}^{5} (cos 5 \times 45^{o} + i sin 5 \times 45^{o}) {(3 \sqrt{2})}^{3} (cos 3 \times 135^{o} + i sin 3 \times 135^{o})}{2^{10} (cos 10 \times + i sin 10 \times 30)}$ $((2sqrt2)^5(cos5 times 45^o + isin5 times45^o)(3sqrt2)^3(cos 3 times 135^o +isin3 times 135^o))/(2^10(cos10 times + isin10 times 30))$

Simplify:

$\frac{128 \sqrt{2} ({cos 225}^{o} + i {sin 225}^{o}) 54 \sqrt{2} ({cos 405}^{o} + i {sin 405}^{o})}{1024 ({cos 300}^{o} + i {sin 300}^{o})}$ $(128sqrt2(cos225^o + isin225^o) 54sqrt2(cos405^o + isin405^o))/(1024(cos 300^o + isin 300^o))$

$\frac{13824 ({cos 630}^{o} + i {sin 630}^{o})}{1024 ({cos 300}^{o} + i {sin 300}^{o})}$ $(13824 (cos630^o + isin630^o)) /(1024(cos300^o + isin300^o))$

Divide:

$\frac{27}{2}$ $27/2$ $({cos 630}^{o} - 300^{o} + i {sin 630}^{o} - 300^{o})$ $(cos630^o-300^o + isin630^o-300^o)$

Simplify:

$\frac{27}{2}$ $27/2$ $({cos 330}^{o} + i {sin 330}^{o})$ $(cos330^o + isin330^o)$

Which gives your answer . . .

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Convert all complex numbers to trigonometric form and then simplify the expression. Write the answer in standard form.? $\frac{{(2 + 2 i)}^{5} {(- 3 + i)}^{3}}{{(\sqrt{3} + i)}^{10}}$ $((2+2i)^5 (-3+i)^3) /(sqrt 3+i)^10$

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Convert all complex numbers to trigonometric form and then simplify the expression. Write the answer in standard form.? ((2+2i)^5 (-3+i)^3) /(sqrt 3+i)^10(2+2i)5(−3+i)3(√3+i)10((2+2i)^5 (-3+i)^3) /(sqrt 3+i)^10

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Convert all complex numbers to trigonometric form and then simplify the expression. Write the answer in standard form.? $\frac{{(2 + 2 i)}^{5} {(- 3 + i)}^{3}}{{(\sqrt{3} + i)}^{10}}$ $((2+2i)^5 (-3+i)^3) /(sqrt 3+i)^10$